Atomic probe Wigner tomography of a nanomechanical system

We propose a scheme to measure the quantum state of a nanomechanical oscillator cooled near its ground state of vibrational motion. This is an extension of the nonlinear atomic homodyning technique scheme first developed to measure the intracavity field in a micromaser. It involves the use of a detector atom that is simultaneously coupled to the resonator via a magnetic interaction and to (classical) optical fields via a Raman transition. We show that the probability for the atom to be found in the ground state is a direct measure of the Wigner characteristic function of the nanomechanical oscillator. We also investigate the back-action effect of this destructive measurement on the state of the resonator.